MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  riotaeqbidv Structured version   Unicode version

Theorem riotaeqbidv 6555
Description: Equality deduction for restricted universal quantifier. (Contributed by NM, 15-Sep-2011.)
Hypotheses
Ref Expression
riotaeqbidv.1  |-  ( ph  ->  A  =  B )
riotaeqbidv.2  |-  ( ph  ->  ( ps  <->  ch )
)
Assertion
Ref Expression
riotaeqbidv  |-  ( ph  ->  ( iota_ x  e.  A ps )  =  ( iota_ x  e.  B ch ) )
Distinct variable group:    ph, x
Allowed substitution hints:    ps( x)    ch( x)    A( x)    B( x)

Proof of Theorem riotaeqbidv
StepHypRef Expression
1 riotaeqbidv.2 . . 3  |-  ( ph  ->  ( ps  <->  ch )
)
21riotabidv 6554 . 2  |-  ( ph  ->  ( iota_ x  e.  A ps )  =  ( iota_ x  e.  A ch ) )
3 riotaeqbidv.1 . . 3  |-  ( ph  ->  A  =  B )
43riotaeqdv 6553 . 2  |-  ( ph  ->  ( iota_ x  e.  A ch )  =  ( iota_ x  e.  B ch ) )
52, 4eqtrd 2470 1  |-  ( ph  ->  ( iota_ x  e.  A ps )  =  ( iota_ x  e.  B ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653   iota_crio 6545
This theorem is referenced by:  dfoi  7483  oieq1  7484  oieq2  7485  ordtypecbv  7489  ordtypelem3  7492  zorn2lem1  8381  zorn2g  8388  cidfval  13906  cidval  13907  cidpropd  13941  lubfval  14440  glbfval  14445  spwval2  14661  spwval  14662  grpinvfval  14848  pj1fval  15331  mpfrcl  19944  evlsval  19945  q1pval  20081  ig1pval  20100  gidval  21806  grpoinvfval  21817  pjhfval  22903  cvmliftlem5  24981  cvmliftlem15  24990  trlfset  31030  dicffval  32045  dicfval  32046  dihffval  32101  dihfval  32102  hvmapffval  32629  hvmapfval  32630  hdmap1fval  32668  hdmapffval  32700  hdmapfval  32701  hgmapfval  32760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-iota 5421  df-fv 5465  df-riota 6552
  Copyright terms: Public domain W3C validator